Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 3 x + p = 0$ and $\gamma$ and $\delta$ be the roots of $x ^ { 2 } - 6 x + q = 0$. If $\alpha , \beta , \gamma , \delta$ from a geometric progression. Then ratio $( 2 q + p ) : ( 2 q - p )$ is
(1) $3 : 1$
(2) $9 : 7$
(3) $5 : 3$
(4) $33 : 31$

The integer $k$, for which the inequality $x ^ { 2 } - 2 ( 3 k - 1 ) x + 8 k ^ { 2 } - 7 > 0$ is valid for every $x$ in $R$ is:
(1) 4
(2) 2
(3) 3
(4) 0

The probability of selecting integers $a \in [ - 5,30 ]$ such that $x ^ { 2 } + 2 ( a + 4 ) x - 5 a + 64 > 0$, for all $x \in R$, is:
(1) $\frac { 7 } { 36 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 4 }$

The coefficients $a , b$ and $c$ of the quadratic equation, $a x ^ { 2 } + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
(1) $\frac { 1 } { 72 }$
(2) $\frac { 1 } { 36 }$
(3) $\frac { 1 } { 54 }$
(4) $\frac { 5 } { 216 }$

Let $a, b \in R$ be such that the equation $ax^2 - 2bx + 15 = 0$ has repeated root $\alpha$ and if $\alpha$ and $\beta$ are the roots of the equation $x^2 - 2bx + 21 = 0$, then $\alpha^2 + \beta^2$ is equal to:
(1) 37
(2) 58
(3) 68
(4) 92

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x + \sqrt { 6 } = 0$ and $\frac { 1 } { \alpha ^ { 2 } } + 1 , \frac { 1 } { \beta ^ { 2 } } + 1$ be the roots of the equation $x ^ { 2 } + a x + b = 0$. Then the roots of the equation $x ^ { 2 } - ( a + b - 2 ) x + ( a + b + 2 ) = 0$ are :
(1) non-real complex numbers
(2) real and both negative
(3) real and both positive
(4) real and exactly one of them is positive

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x ^ { 2 } + \alpha x + \beta > 0$, for all $x \in R$, is
(1) $\frac { 17 } { 36 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 19 } { 36 }$

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - 4 \lambda x + 5 = 0$ and $\alpha , \gamma$ be the roots of the equation $x ^ { 2 } - ( 3 \sqrt { 2 } + 2 \sqrt { 3 } ) x + 7 + 3 \lambda \sqrt { 3 } = 0$. If $\beta + \gamma = 3 \sqrt { 2 }$, then $( \alpha + 2 \beta + \gamma ) ^ { 2 }$ is equal to

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x ^ { 2 } - 12 x + [ x ] + 31 = 0$ and $x ^ { 2 } - 5 | x + 2 | - 4 = 0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^ { 2 } + m n + n ^ { 2 }$ is equal to

If the value of real number $\alpha > 0$ for which $x^{2} - 5\alpha x + 1 = 0$ and $x^{2} - \alpha x - 5 = 0$ have a common real roots is $\frac{3}{\sqrt{2\beta}}$ then $\beta$ is equal to $\_\_\_\_$

For $0 < c < b < a$, let $( a + b - 2 c ) x ^ { 2 } + ( b + c - 2 a ) x + ( c + a - 2 b ) = 0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in ( -1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$.
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true

Let $S = \left\{ \sin ^ { 2 } 2 \theta : \left( \sin ^ { 4 } \theta + \cos ^ { 4 } \theta \right) x ^ { 2 } + ( \sin 2 \theta ) x + \left( \sin ^ { 6 } \theta + \cos ^ { 6 } \theta \right) = 0 \right.$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3 \left( ( \alpha - 2 ) ^ { 2 } + ( \beta - 1 ) ^ { 2 } \right)$ equals $\_\_\_\_$

If the set of all $\mathrm { a } \in \mathbf { R }$, for which the equation $2 x ^ { 2 } + ( a - 5 ) x + 15 = 3 \mathrm { a }$ has no real root, is the interval $( \alpha , \beta )$, and $X = \{ x \in Z : \alpha < x < \beta \}$, then $\sum _ { x \in X } x ^ { 2 }$ is equal to :
(1) 2109
(2) 2129
(3) 2119
(4) 2139

Let $\alpha _ { \theta }$ and $\beta _ { \theta }$ be the distinct roots of $2 x ^ { 2 } + ( \cos \theta ) x - 1 = 0 , \theta \in ( 0,2 \pi )$. If m and M are the minimum and the maximum values of $\alpha _ { \theta } ^ { 4 } + \beta _ { \theta } ^ { 4 }$, then $16 ( M + m )$ equals :
(1) 24
(2) 25
(3) 17
(4) 27

If the equation $\mathrm { a } ( \mathrm { b} - \mathrm { c } ) \mathrm { x } ^ { 2 } + \mathrm { b } ( \mathrm { c } - \mathrm { a } ) \mathrm { x } + \mathrm { c } ( \mathrm { a } - \mathrm { b } ) = 0$ has equal roots, where $\mathrm { a } + \mathrm { c } = 15$ and $\mathrm { b } = \frac { 36 } { 5 }$, then $a ^ { 2 } + c ^ { 2 }$ is equal to

Q80. The coefficients $a , b , c$ in the quadratic equation $a x ^ { 2 } + b x + c = 0$ are chosen from the set $\{ 1,2,3,4,5,6,7,8 \}$ . The probability of this equation having repeated roots is :
(1) $\frac { 1 } { 128 }$
(2) $\frac { 1 } { 64 }$
(3) $\frac { 3 } { 256 }$
(4) $\frac { 3 } { 128 }$

Q82. Let $S = \left\{ \sin ^ { 2 } 2 \theta : \left( \sin ^ { 4 } \theta + \cos ^ { 4 } \theta \right) x ^ { 2 } + ( \sin 2 \theta ) x + \left( \sin ^ { 6 } \theta + \cos ^ { 6 } \theta \right) = 0 \right.$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3 \left( ( \alpha - 2 ) ^ { 2 } + ( \beta - 1 ) ^ { 2 } \right)$ equals $\_\_\_\_$

Q90. Let $\mathrm { a } , \mathrm { b }$ and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $a x ^ { 2 } + b x + c = 0$ has all real roots is $\frac { m } { n } , \operatorname { gcd } ( \mathrm {~m} , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$

ANSWER KEYS

1. (2)	2. (2)
9. (3)	10. (1)
17. (1)	18. (2)
25. (36)	26. (25)
33. (2)	34. (2)
41. (1)	42. (2)
49. (3)	50. (1)
$57 . ( 8 )$	58. (2)
65. (1)	66. (4)
73. (2)	74. (3)
81. (9)	82. (1)
89. (39)	90. (19)

1. (4)
2. (2)
3. (3)
4. (160)
5. (1)
6. (1)
7. (164)
8. (3)
9. (1)
10. (1)
11. (9)
12. (4)
13. (2)
14. (3)
15. (10)
16. (4)
17. (2)
18. (4)
19. (4)
20. (3)
21. (3)
22. (32)
23. (2)
24. (2)
25. (3)
26. (200)
27. (1)
28. (4)
29. (2)
30. (3)
31. (2)
32. (2)
33. (25)
34. (4)
35. (2)
36. (100)
37. (15)
38. (1)
39. (2)
40. (82)
41. (2)
42. (1)
43. (4)
44. (14)
45. (2)
46. (3)
47. (1)
48. (3)
49. (20)
50. (17)
51. (4)
52. (3)
53. (4)
54. (3)
55. (4)
56. (4)
57. (4)
58. (3)
59. (2)
60. (3)
61. (3)
62. (1)
63. (1)
64. (2)
65. (1010)
66. (81)

Let $\alpha , \beta$ be the roots of quadratic equation $12 \mathrm { x } ^ { 2 } - 20 \mathrm { x } + 3 \lambda = 0$, $\lambda \in \mathbf { z }$. If $1 / 2 \leq | \beta - \alpha | \leq 3 / 2$ then the sum of all possible valued of $\lambda$ is $\_\_\_\_$ -

Let the equation $\mathrm { x } ^ { 4 } - \mathrm { ax } ^ { 2 } + 9 = 0$ have four real and distinct roots.
Then the least integral value of $a$ is
(A) 5
(B) 7
(C) 6
(D) 8

Consider the following quadratic equations in $x$
$$x^2+2x-15=0 \tag{1}$$ $$2x^2+3x+a^2+12a=0 \tag{2}$$
Let us denote the two solutions of (1) by $\alpha$ and $\beta$ ($\alpha < \beta$). We are to find the range of values which $a$ in (2) can take, in order that (2) has two real solutions $\gamma$ and $\delta$ and they satisfy
$$\alpha < \gamma < \beta < \delta.$$
(1) $\alpha = \mathbf{AB}$ and $\beta = \mathbf{C}$.
(2) When we set $b = a^2+12a$, from the condition $\alpha < \gamma$ we have
$$b > \mathbf{DEF}$$
and from the condition $\gamma < \beta < \delta$ we have
$$b < \mathbf{GHI}.$$
Hence the range of the values which $a$ can take is
$$\mathbf{JK} < a < \mathbf{LM}, \quad \mathbf{NO} < a < \mathbf{PQ},$$
where $\mathrm{JK} < \mathrm{NO}$.

Let $a$ be a constant. For the two functions in $x$
$$\begin{aligned} & f ( x ) = 2 x ^ { 2 } + x + a - 2 \\ & g ( x ) = - 4 x - 5 \end{aligned}$$
we are to find the real values of $x$ for which $f ( x ) = g ( x )$ and also find the values of the two functions there.
(1) For each of $\mathbf { N } , \mathbf { O }$ and $\mathbf { P }$ in the following statements, choose the appropriate condition from (0) $\sim$ (8) below.
When $\mathbf { N }$, there are two real values of $x$ for which $f ( x ) = g ( x )$. When $\mathbf { O }$, there is only one real value of $x$ for which $f ( x ) = g ( x )$. When $\mathbf{P}$, there is no real value of $x$ for which $f ( x ) = g ( x )$.
(0) $a > \frac { 1 } { 8 }$
(1) $a = \frac { 17 } { 8 }$
(2) $a = \frac { 1 } { 6 }$
(3) $a < \frac { 1 } { 6 }$
(4) $a < \frac { 17 } { 8 }$
(5) $a < \frac { 1 } { 8 }$ (6) $a > \frac { 1 } { 6 }$ (7) $a = \frac { 1 } { 8 }$ (8) $a > \frac { 17 } { 8 }$
(2) When N, the values of $x$ for which $f ( x ) = g ( x )$ are $\frac { - \mathrm { Q } \pm \sqrt { \mathrm { R } - \mathbf { S } a } } { \mathbf{T} }$, and the values of the functions there are $\mp \sqrt { \mathbf { U } - \mathbf { V } a }$.
When O, the value of $x$ for which $f ( x ) = g ( x )$ is $- \frac { \mathrm { W } } { \mathrm { X} }$, and the value of the functions there is $\mathbf{Y}$.
(3) Consider the case where $f ( x ) = g ( x )$ and the absolute value of these functions there is greater than or equal to 3. The condition for this case is that $a \leqq - \mathbf { Z }$.

Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.

Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.

Let $a$ be a real number. Consider the quadratic expressions in $x$
$$\begin{aligned} & A = x^2 + ax + 1 \\ & B = x^2 + (a+3)x + 4 \end{aligned}$$
(1) The range of values taken by $a$ such that there exists a real number $x$ satisfying $A + B = 0$ is
$$a \leq -\sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \text{ or } \sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \leq a.$$
(2) The range of values taken by $a$ such that there exists a real number $x$ satisfying $AB = 0$ is
$$a \leq \mathbf{EF} \text{ or } \mathbf{G} \leq a.$$
(3) There exists a real number $x$ satisfying $A^2 + B^2 = 0$ only when $a = \mathbf{H}$. In this case $x = \mathbf{IJ}$.